67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.

f(x) = x^2/3, for x>0

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First, understand that if y = f(x) is a function, then its inverse function is denoted as x = f^(-1)(y). We need to find the derivative of this inverse function.

Given the function f(x) = x^(2/3), we need to find the inverse function. To do this, set y = x^(2/3) and solve for x in terms of y.

Raise both sides to the power of 3/2 to solve for x: x = y^(3/2). This gives us the inverse function f^(-1)(y) = y^(3/2).

To find the derivative of the inverse function, use the formula: (d/dx)[f^(-1)(y)] = 1 / (f'(f^(-1)(y))).

Calculate f'(x) for the original function: f'(x) = (2/3)x^(-1/3). Substitute f^(-1)(y) = y^(3/2) into this derivative to find the derivative of the inverse function.

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